Is it possible for a NN to reach the same results as CNNs?
NNs won't be able to reach the performance of CNNs, in general.

By a Neural Network, I am assuming you're pointing to 'vanilla' vector based neural networks. 

Let's take the MNIST dataset, it performs almost similar in both NNs and CNNs; the reason being digits of almost same size, and similar spatial drawings. To put it easy, all the digits roughly take up the same area as the other 60k. If you were to zoom in our zoom out these digits, the trained NN may not perform well.

Why?

Plain NNs lack the ability to extract 'position independent' features. A cat is a cat to your eye, no matter whether you saw it in the center of an image, or left corner. CNNs use 'filters' to extract 'templates' of cat. Hence CNNs can 'localize' what it is searching for. 

You could say, NNs look at the 'whole' data to make a sense, CNNs look at the 'special' features at certain parts to make a sense.

This is also a reason why CNNs are popular. NNs are suited for their own applications. Every input node in a vectorized Neural Networks represent a feature, and that feature is tied to a purpose throughout training, i.e. its position is fixed throughout the training.